So far, I've discussed rules that take place within a single group — a row, column, or box. All of these rules can be summarized as follows:
- If N cells together contain exactly N values, remove those values from the other cells of the group
- If N values can only appear in exactly N calls, remove the other values from those cells
When N is 1, the first rule is the rule of elimination; when N is 2, it's the rule of pairs, when N is three it's the rule of (interlocking) triples, etc.
When N is 1, the second rule is the rule of uniqueness. I don‘t have a good name for the second rule when N is larger than 1. Any suggestions? (Someone suggested biniques.)
Now we are ready to look at rules that look outside a single group, and deal with the interactions between groups. In particular, today's rule deals with the interaction between a box and the rows and columns that intersect it.
Consider the lower-right box (box 9) in the grid below. I've outlined the right part of the box in red. Notice that the only places that can contain a 2 are along the right side of the box. While we don‘t know which cell of the box actually contains the 2, we can be certain that it will be one of the cells in column 9. So, we can remove 2's from the rest of column 9 (the blue rectangle).
Similarly, the only places in cell 3 that can hold a 7 are in column 7 (inside the green oval). So, we can remove the rest of the 7's in that column (inside the orange rectangle).
Some Sudoku enthusiasts refer to this technique as “2 in a bed“ and “3 in a bed“ (for the case where there are three possible cells for a value within a box, all in the same row or column).
The rule can be reversed: if there are only two or three places in a row (or column) that can contain a value, and those places all live inside the same box, the value may be removed from the rest of the box.
Here‘s an example of a “reverse bed”:
In row 5 (highlighted in red), there are two cells (circled) that can hold a 3, and both happen to be in box 6. This allows us to remove the 3 from the rest of box 6 — cells (7,6) and (9,6), circled in blue.
Here are some puzzles that use “beds“ and “reverse beds”: